To find the units digit of the sum $2013^1 + 2013^2 + 2013^3 + \dots + 2013^{2013}$, we focus on the units digit of $2013^n$ for each $n$.

Step 1: Identify the units digit of $2013^n$

The units digit of $2013^n$ is determined by the units digit of $3^n$ because the other digits do not affect the units digit.

Step 2: Determine the pattern in the units digits of powers of 3

Let's compute the units digits of the first few powers of 3:

• $3^1 = 3$ (units digit = 3)
• $3^2 = 9$ (units digit = 9)
• $3^3 = 27$ (units digit = 7)
• $3^4 = 81$ (units digit = 1)
• $3^5 = 243$ (units digit = 3)

Notice that the units digits repeat every 4 powers: 3, 9, 7, 1.

Step 3: Apply the pattern to find the units digit of each term

Since the pattern repeats every 4 terms, we calculate the remainder when the exponent is divided by 4 to determine the corresponding units digit:

• If $n \equiv 1 \ (\text{mod} \ 4)$, the units digit of $2013^n$ is 3.
• If $n \equiv 2 \ (\text{mod} \ 4)$, the units digit of $2013^n$ is 9.
• If $n \equiv 3 \ (\text{mod} \ 4)$, the units digit of $2013^n$ is 7.
• If $n \equiv 0 \ (\text{mod} \ 4)$, the units digit of $2013^n$ is 1.

Step 4: Find the sum of the units digits from $2013^1$ to $2013^{2013}$

We now add the units digits for the exponents from 1 to 2013. Since 2013 is not a multiple of 4, we calculate how many complete cycles of 4 occur in the sum and then add the remainder.

• $2013 \div 4 = 503$ complete cycles, with a remainder of 1.

Each cycle contributes a units digit sum of $3 + 9 + 7 + 1 = 20$.

Therefore, the sum of the units digits is: $503 \times 20 + 3 = 10060 + 3 = 10063$ The units digit of 10063 is 3.

Final Answer:

The units digit of the entire sum $2013^1 + 2013^2 + \dots + 2013^{2013}$ is 3.

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Would you like further explanation on any part? Here are some related questions you might find interesting:

1. What is the units digit of $7^{1234}$?
2. How does modular arithmetic help in finding units digits?
3. What is the units digit of $2^{50}$?
4. Can this method be applied to other bases, such as finding the units digit of $2017^n$?
5. How would you calculate the units digit of large sums like $123^1 + 123^2 + \dots + 123^{1000}$?

Tip: When working with powers and units digits, always look for repeating patterns in the sequence to simplify your calculations.